The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calculation of flow variables in case of flow in partially full conduits, as they also possess a free surface like that of open channel flow. All flow in so-called open channels is driven by gravity.
It was first presented by the French engineer in 1867, and later re-developed by the Irish engineer Robert Manning in 1890. Thus, the formula is also known in Europe as the Gauckler–Manning formula or Gauckler–Manning–Strickler formula (after ).
The Gauckler–Manning formula is used to estimate the average velocity of water flowing in an open channel in locations where it is not practical to construct a weir or flume to measure flow with greater accuracy. Manning's equation is also commonly used as part of a numerical step method, such as the standard step method, for delineating the free surface profile of water flowing in an open channel.
Formulation
The Gauckler–Manning formula states:
where: is the cross-sectional average velocity (L/T; ft/s, m/s); is the Gauckler–Manning coefficient. Units of are often omitted, however is not dimensionless, having units of: (T/[L1/3]; s/[m1/3]). is the hydraulic radius (L; ft, m); is the stream slope or hydraulic gradient, the linear hydraulic head loss loss (L/L); it is the same as the channel bed slope when the water depth is constant. (). is a conversion factor between SI and English units. It can be left off, as long as you make sure to note and correct the units in the term. If you leave in the traditional SI units, is just the dimensional analysis to convert to English. for SI units, and for English units. (Note: (1 m)1/3/s = (3.2808399 ft)1/3/s = 1.4859 ft1/3/s)
NOTE: strickler = 1/ manning. The coefficient strickler varies from 20 (rough stone and rough surface) to 80 m1/3/s (smooth concrete and cast iron).
The discharge formula, , can be used to rewrite Gauckler–Manning's equation by substitution for . Solving for then allows an estimate of the volumetric flow rate (discharge) without knowing the limiting or actual flow velocity.
The formula can be obtained by use of dimensional analysis. In the 2000s this formula was derived theoretically using the phenomenological theory of turbulence.
Hydraulic radius
The hydraulic radius is one of the properties of a channel that controls water discharge. It also determines how much work the channel can do, for example, in moving sediment. All else equal, a river with a larger hydraulic radius will have a higher flow velocity, and also a larger cross sectional area through which that faster water can travel. This means the greater the hydraulic radius, the larger volume of water the channel can carry.
Based on the 'constant shear stress at the boundary' assumption, hydraulic radius is defined as the ratio of the channel's cross-sectional area of the flow to its wetted perimeter (the portion of the cross-section's perimeter that is "wet"):
where: is the hydraulic radius (L); is the cross sectional area of flow (L2); is the wetted perimeter (L).
For channels of a given width, the hydraulic radius is greater for deeper channels. In wide rectangular channels, the hydraulic radius is approximated by the flow depth.
The hydraulic radius is not half the hydraulic diameter as the name may suggest, but one quarter in the case of a full pipe. It is a function of the shape of the pipe, channel, or river in which the water is flowing.
Hydraulic radius is also important in determining a channel's efficiency (its ability to move water and sediment), and is one of the properties used by water engineers to assess the channel's capacity.
Gauckler–Manning coefficient
The Gauckler–Manning coefficient, often denoted as , is an empirically derived coefficient, which is dependent on many factors, including surface roughness and sinuosity. When field inspection is not possible, the best method to determine is to use photographs of river channels where has been determined using Gauckler–Manning's formula.
The friction coefficients across weirs and orifices are less subjective than along a natural (earthen, stone or vegetated) channel reach. Cross sectional area, as well as , will likely vary along a natural channel. Accordingly, more error is expected in estimating the average velocity by assuming a Manning's , than by direct sampling (i.e., with a current flowmeter), or measuring it across weirs, flumes or orifices.